Foundations — Conservation of linear momentum — derivation from Newton's third law

1.4.3 · D1 · Physics › Momentum & Collisions › Conservation of linear momentum — derivation from Newton's t

Is page mein kuch bhi assumed nahi hai. Agar parent note mein koi symbol likha tha, toh hum use yahan se ground up banate hain — ek aisi order mein jahan har idea apne pehle wale idea par khada ho. Upar se neeche padho.

1. Ek number jo point bhi kare — arrow (vector)

Plain speed "5 metres per second" sirf ek number hai — yeh nahi batata kidhar. Velocity kehta hai "5 m/s daayein taraf". Yeh extra "kidhar" hi woh cheez hai jo upar wala chhota arrow, $\vec{v}$, promise karta hai.

Tasveer: ek arrow zameen par draw kiya hua. Orange arrow dekho — uski length "kitni speed" hai, uska pointing "kidhar" hai. Chhota arrow = slow, lamba arrow = fast.

Topic ko iske zaroorat kyun hai: momentum bhi ek arrow hoga, kyunki ek bullet kis direction mein jaati hai yeh utna hi important hai jitni uski speed. Agar hum direction hata dete, toh hum yeh kabhi explain nahi kar sakte ki gun *peeche kyun kickback karti hai.

2. Arrow ko $x$ aur $y$ mein todna (components)

$\hat{x}$ ko "x-hat" padhte hain. Hat ka matlab hai "yeh ek pure direction hai, length ek, koi size baggage nahi." Toh $6\hat{x}$ ka matlab hai "6 units, daayein taraf point karta hua."

Tasveer: magenta arrow woh shadow hai jo vector zameen par dalta hai ($v_x$); violet arrow woh shadow hai jo woh wall par dalta hai ($v_y$). Dono shadows milke poora arrow dobara bana dete hain.

Topic ko iske zaroorat kyun hai: "har axis ko independently conserve karo" wala trick (parent Example 3) components ke bina impossible hai.

3. Mass $m$ — kitna stuff hai

Tasveer: do balls socho — ek marble aur ek bowling ball. Same push dene par marble uchhal jaata hai, bowling ball muskil se hilta hai — woh reluctance hi mass hai.

Topic ko iske zaroorat kyun hai: momentum mass times velocity hoga. Mass woh number hai jo batata hai "ek given speed par kitna motion-stuff worth hai."

4. Momentum $\vec{p}=m\vec{v}$ — "motion-money"

Tasveer: same velocity arrow, lekin ek bhaari object ke liye woh stretch hokar ek lamba momentum arrow ban jaata hai. Ek slow truck ek fast bicycle se zyada momentum carry kar sakta hai kyunki mass arrow ko stretch karta hai.

"Linear" kyun? Iska sirf yeh matlab hai ki straight-line momentum — ise spinning (angular) momentum se alag karne ke liye use karte hain, jo parent topic mein nahi aata. Hamare liye "momentum" aur "linear momentum" ek hi cheez hain.

Topic ko iske zaroorat kyun hai: yahi woh quantity hai jiske baare mein poora chapter keh raha hai ki conserve hoti hai. Yeh closed-room analogy mein "money" hai.

5. Greek $\Sigma$ — "sab add karo" (sum)

Toh $\sum _i{m}_i{\vec{v}}_i$ ka matlab hai: particle 1 ka momentum lo, plus particle 2 ka, plus particle 3 ka, … aakhir tak, aur arrows add karo.

Topic ko iske zaroorat kyun hai: ek system ka "total momentum" matlab hai har piece ke arrow ka sum. Parent ka $\vec{P}={\sum }_i{\vec{p}}_i$ literally yahi command hai.

6. Force $\vec{F}$ — ek push ya pull (yeh bhi ek arrow hai)

Tasveer: ek haath box ko dhakelta hua — arrow box par shuru hota hai (jahan feel hota hai) aur dhakele ki direction mein point karta hai.

Topic ko iske zaroorat kyun hai: poori derivation is baare mein hai ki forces momentum ko kaise badal dete hain, aur kaise ek push ke dono forces cancel ho jaate hain.

7. Rate-of-change ka idea $\frac{d}{dt}$ — derivative

Yeh genuinely naya machinery ka ek piece hai, isliye hum dheere jaate hain.

Tasveer: momentum ka time ke saath ek curve. Derivative us curve ki steepness (slope) hai ek point par — steep matlab tezi se badal raha hai, flat matlab bilkul nahi badal raha.

"$d$" kyun, plain subtraction sign kyun nahi? Kyunki $d/dt$ instant-by-instant changing force ko handle karta hai, jabki plain "after minus before" ke liye force constant rehni chahiye. Newton's second law honestly $\vec{F}=d\vec{p}/dt$ hai; jaana-maana $\vec{F}=m\vec{a}$ sirf woh special case hai jab $m$ nahi badlta.

Topic ko iske zaroorat kyun hai: poora proof isi par chalta hai ki "total momentum ka derivative zero hai, isliye total momentum constant hai." Derivative ke idea ke bina, parent ke Steps 3–5 ka koi matlab nahi.

8. Before vs after — prime mark ${}^{\prime }$

Topic ko iske zaroorat kyun hai: conservation ek aisa statement hai jo pehle aur baad ko compare karta hai — $m_1{\vec{v}}_1+m_2{\vec{v}}_2=m_1{\vec{v}}_1^{\prime }+m_2{\vec{v}}_2^{\prime }$ — aur prime hi woh tarika hai jis se hum dono moments ko alag karte hain.

9. "Isolated" aur "external" — fine print

Tasveer: jo objects tumhe concern karte hain unke around ek dotted circle draw karo. Jo arrows circle cross karte hain woh external hain; jo arrows andar hi hain woh internal hain.

Topic ko iske zaroorat kyun hai: momentum sirf ek isolated system ke liye conserved hota hai. Internal pushes Newton's third law se cancel ho jaate hain; sirf external pushes total ko badal sakte hain.

Prerequisite map

Equipment checklist

Khud test karo — daayein side cover karo aur zawaab zor se bolo.

Connections

• Yeh foundations Hinglish mein
• Newton's Third Law — internal-force pairs jo cancel hote hain woh supply karta hai
• Newton's Second Law (momentum form) — jahan $\vec{F}=d\vec{p}/dt$ rehta hai
• Impulse–Momentum Theorem — force ko instantaneously ki jagah time ke saath add karta hai
• Centre of Mass Motion — summed momentum $\vec{P}=M{\vec{v}}_{cm}$
• Elastic vs Inelastic Collisions — energy compare karne ke liye inhi symbols ka use karta hai
• Rocket Equation — momentum bookkeeping jab mass khud badal jaata hai